cs355 - theory of computation lecture 2: mathematical preliminaries

CS355 - Theory of Computation

Lecture 2: Mathematical Preliminaries

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth

Set theory

• A set is a collection of objects.• These objects are referred to as its

elements.• The order of these elements is not

important.• The size of a set is its cardinality - | |

• Natural number when the set is finite when the set is infinite• Empty set has cardinality of zero - |ø| = zero.

Membership

• Set membership is denoted using and non membership by – a {a , b} and c {a , b}– apple {apple, pear, banana}– apple {apples, pears, bananas}

Subsets

• A set A whose elements are also elements of a set B is called a subset of B

• A B• A = {0,1,2} and B = {0,2,3,4,1} then A B

• If |B| > |A| then A is known as a proper subset of B, denoted by A B.

• Note, A A and ø A for all sets A.• If A = B then A B and B A.

Some Elementary Logic

• and• or• implies/only if• not• universal qualifier (for all)• existential qualifier (there exists)

More Set Theory

• union - all the elements of both sets but no duplicates

• intersection - all the elements that are in both sets • - difference (A – B) - the set of elements that are in A

but not in B • symmetric difference - the set of elements belonging

to one but not both of two given sets. It is therefore the union of the complement of A with respect to B and B with respect to A, and corresponds to the XOR operation in Boolean logic.

Power Set

• The power set 2n is the set of all possible subsets of A including ø and A itself.

• If A is finite and consists of n elements, then the power set has 2n elements.

• e.g. if A = {a,b} then 2A = {ø, {a}, {b}, {a,b}}

• e.g. if A = {a,b,c} then 2A = {ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

Relations

• an ordered pair of elements• It differs from {a,b} in that:

• The order of the elements if important• The same element may occur twice

Cartesian Product

• The Cartesian product (A×B) is the set of all ordered pairs <x,y> with x A and y B

• Therefore if A = {0,1,2} and B = {c,d} then

A×B = {<0,c>,<0,d>, <1,c>,<1,d>, <2,c>,<2,d>}

B×A = {<c,0>,<c,1>, <c,2>,<d,0>, <d,1>,<d,2>}

Binary Relation

• Any subset (call it R) of A×B is called a binary relation between A and B

– R = {<0 , c>,<1 , d>} is a binary relation between A and B (A = {0, 1, 2} and B = {c, d})

– Note that the singular ‘binary relation’ relates to a set of ordered pairs

Predicate

• The predicate R(a,b) is true if <a,b> R

• In future when we talk about a relation R we mean the combination of the set R and its implicit predicate R()

• a ^ b where the predicate R(a,b) = trueThe domain = {a:a} andThe codomain (range) = {b:b}

• If we have more than one ordered pair <x1,x2,…,xn> we call it an ordered n-tuple.

Functions

• A function f: A → B is a binary relation between A and B where only one tuple <a,b> exists for each a A

• f may be written f(a) = b

Functions

• Given a function f(a) = b we can say:– The value of f is a in b– a is the argument and b is the value– f is injective if there is a unique b B for every a A– f is surjective if every b B has an a A– f is bijective if it is both injective and surjective

Recap: Number Sequences

• Natural numbers (N): the whole positive numbers including zero: 0, 1, 2, 3,....

• Integers numbers (Z): the natural numbers plus their negative counterparts: …,-2,-1,0,1,2,…

• Rational numbers (Q): the integers plus the rational fractions - those that can be expressed as the ratio of two integers: ⅓,⅜,…

• Real numbers (R): the rational numbers plus the irrational numbers: 0.000123, 1.24,…

Cardinality and Infinite Sets

• The Cardinality is the number of elements of a finite set - |A|

• What about infinite sets?

• An infinite set contains infinitely many elements – can’t write a list of all the elements of such sets– Use “…” to mean “continue sequence forever”– Set of Natural Numbers: {1,2,3,…}

• Two infinite sets, A and B are equinumerous/equivalent (A≡B) if a bijection exists between them.– Note, we don’t have to be able to compute it in

every instance, just be certain one exists.

• Therefore |A| = |B| A≡B• Putting two infinite sets into one-to-one

correspondence is an infinite task, and we don't pretend that we can do it (that is, finish it) in finite time.

• To show that an infinite set, like the even numbers, can be put into one-to-one correspondence with another, like the odd numbers, we need only produce a rule-governed sequence for each set which runs through the members without omission or repetition:– for example, 2, 4, 6... and 1, 3, 5....

• If we can do so, then we know that the nth term of one sequence will have a counterpart in the nth term of the other, and vice versa, guaranteeing one-to-one correspondence for each element.

• However there are some infinite sets which are not equinumerous. We use this to prove that uncomputable numbers exist.

• A set is countable iff (if and only if) its cardinality is either finite or equal to N (the Natural numbers) – i.e. if a bijection exists with a subset of the natural numbers - A≡N.

• Other ways to identify a countable set:– May one order the set such that between any

two elements a, b there is a finite number of elements?, or

– Could one write out the first two elements of the set?

• This is possible with N and Z, but not with R.

• Finite sets are countable:– Example: Real numbers with two decimal

places of accuracy between 0 and 1 is a countable set.

• If a set is not countable it is uncountable.

Graphs

• A graph is a set of points with lines connecting some of the points.

• The points are known as nodes or vertices.• The connecting lines are known as edges – the

number of edges = degree (each node has degree 2 and degree 3, respectively below)

Graphs

• A path in a graph is a sequence of nodes connected by edges.

• A graph is connected if, for every two distinct nodes a and b, there is a path from a to b.

• A cycle is a path within a graph that starts and ends at the same node.

• A graph is a tree if it is connected and has no simple cycles – may contain a specially designated node called the root.

• Nodes of degree 1 in a tree, apart from the root, are called the leaves of the tree.

Graphs

Leaves of tree

path cycle

Directed Graph• If a graph has arrows instead of lines it is a

directed graph.

• The number of arrows leaving a node is the outdegree of that node and the number of arrows entering a node is its indegree.

Directed Graph

• In a directed graph an edge from node i to node j is represented as a pair (i, j)

• The formal description of a directed graph G is (V, E) where V is the set of nodes and E the set of edges.

• ({1,2,3,4,5,6}, {(1,2), (1,5), (2,1), (2,4), (5,4), (5,6), (6,1), (6,3)})

Graphs

• In a binary tree each node which is not a leaf has at most two children.

• If we distinguish between left and right children then the tree is ordered.

• In a complete tree each node which is not a leaf has exactly two children.

• A complete tree is perfect if all leaves have the same height.

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